Exactly solvable spin IsingHeisenberg diamond chain with the secondneighbor interaction between nodal spins
Abstract
The spin1 IsingHeisenberg diamond chain with the secondneighbor interaction between the nodal spins is rigorously solved using the transfermatrix method. Exact results for the ground state, magnetization process and specific heat are presented and discussed in particular. It is shown that the furtherneighbor interaction between the nodal spins gives rise to three novel ground states with a translationally broken symmetry, but at the same time, it does not increases the total number of intermediate plateaus in a zerotemperature magnetization curve compared with the simplified model without this interaction term. The zerofield specific heat displays interesting thermal dependencies with a single or doublepeak structure.
pacs:
05.50.+q, 75.10.Hk, 75.10.Jm, 75.10.Pq, 75.40.Cx1 Introduction
The spin1/2 IsingHeisenberg diamond chain has received considerable research interest since Čanová et al. [1, 2] reported on first exact results for this interesting quantum spin chain. Early exact results for the spin1/2 IsingHeisenberg diamond chain have predicted a lot of intriguing magnetic features such as an intermediate onethird magnetization plateau or doublepeak temperature dependencies of specific heat and susceptibility [1, 2]. In spite of a certain oversimplification, the generalized version of the spin1/2 IsingHeisenberg diamond chain qualitatively reproduces magnetization, specific heat and susceptibility data reported on the azurite Cu(CO)(OH), which represents the most prominent experimental realization of the spin1/2 diamond chain [3, 4, 5, 6]. A lot of attention has been therefore paid to a comprehensive analysis of quantum and thermal entanglement [7, 8, 9, 10, 11], correlation functions [12], Lyapunov exponent [13], zeros of partition function [14], magnetocaloric effect [15], the influence of asymmetric [16, 17], furtherneighbor [18] and fourspin interactions [19, 20].
Recently, it has been verified that the spin1 IsingHeisenberg diamond chain may display more diverse ground states and magnetization curves than its spin1/2 counterpart [21, 22]. It actually turns out that the magnetization curve of the spin1 IsingHeisenberg diamond chain involves intermediate plateaus at zero, onethird and twothirds of the saturation magnetization even if the relevant ground states do not have translationally broken symmetry [21, 22], while the spin1/2 IsingHeisenberg diamond chain may involve those intermediate plateaus just if the asymmetric, furtherneighbor [18] and/or fourspin interactions [19, 20] break a translational symmetry of the relevant ground states. In the present work, we aim to examine the role of the secondneighbor interaction between the nodal spins on the ground state and magnetization process of the spin1 IsingHeisenberg diamond chain.
The paper is organized as follows. In Sec. 2 we will introduce the investigated spinchain model and briefly describe basic steps of our rigorous calculation. The most interesting results for the ground state, magnetization process and specific heat are discussed in detail in Sec. 3. Finally, some conclusions and future outlooks are briefly mentioned in Sec. 4.
2 The model and its exact solution
We consider the spin IsingHeisenberg model on a diamond chain in a presence of the external magnetic field. The primitive unit cell of a diamond chain consists of two Heisenberg spins and , which interact symmetrically via Isingtype interaction with two nearestneighbor Ising spins and (see Fig. 1). The total Hamiltonian of the model under investigation may be represented as a sum over block Hamiltonians , where
(1)  
In above, and () denote spatial components of the spin operators, stands for the Ising spin, labels the XXZ interaction between the nearestneighbor Heisenberg spins, is a spatial anisotropy in this interaction, is the Ising interaction between the nearestneighbor Ising and Heisenberg spins and finally, the last two terms and determine Zeeman’s energy of the Heisenberg and Ising spins in a longitudinal magnetic field.
The important part of our further calculations is based on the commutation relation between different block Hamiltonians , which will allow us to partially factorize the partition function of the model and represent it as a product over block partition functions
(2) 
where , is Boltzmann’s constant, is the absolute temperature, marks a summation over spin states of all Ising spins and means a trace over the spin degrees of freedom of two Heisenberg spins from the th block. After a straightforward diagonalization of the cell Hamiltonian (1), which corresponds to the spin1 quantum Heisenberg dimer in a magnetic field, one obtains the following expressions for the respective eigenvalues
(3) 
where and . Now, one may simply perform a trace over the spin degrees of freedom of the spin1 Heisenberg dimers on the righthandside of Eq. (2) and the partition function can be consequently rewritten into the following form
(4) 
where the expression can be viewed the standard transfer matrix
(5) 
Here, the subscripts and 0 denote three available spin states of the Ising spins and involved in the transfer matrix (5), which has precisely the same form as the transfer matrix of the generalized spin1 BlumeEmeryGriffiths chain diagonalized in Refs. [23, 24, 25]. Of course, individual elements of the transfer matrix (5) are defined through the formula
(6) 
which includes the set of eigenvalues (3) for the spin1 quantum Heisenberg dimer and can be rewritten as
(7) 
With regard to Eq. (4), the partition function of the spin1 IsingHeisenberg diamond chain can be expressed through three eigenvalues of the transfer matrix [23, 24]
(8) 
which can be evaluated from
(9) 
The coefficients , , , and entering the eigenvalues (9) are given by
Next, let us denote the largest transfermatrix eigenvalue , because the contribution of two smaller transfermatrix eigenvalues to the partition function may be completely neglected in the thermodynamic limit
(10) 
The free energy per elementary diamond cell can be obtained from the largest eigenvalue of the transfer matrix (6) according to the formula
(11) 
Knowledge of the free energy allows us to obtain thermodynamic quantities of the system (such as entropy, magnetization, specific heat, etc.) in terms of the free energy and/or its derivatives. In particular, for the entropy and the heat capacity per unit cell one can obtain
(12) 
One may also obtain the singlesite magnetization of the Ising () and the Heisenberg () spins, which are given by
(13) 
Finally, the total magnetization per site follows from
(14) 
3 Results and discussion
In this section, we will investigate in detail the ground state, magnetization process and specific heat of the spin1 IsingHeisenberg diamond chain with antiferromagnetic coupling constants and . Hereafter, we will consider for simplicity the uniform external magnetic field acting on the Ising and Heisenberg spins . Again for simplicity, the Boltzmann’s constant is set to unity and a strength of the Ising coupling will be subsequently used for introducing a set of dimensionless parameters
(15) 
which determine a relative strength of the Heisenberg coupling, the secondneighbor interaction between the nodal spins, the magnetic field and temperature, respectively. The spin1 IsingHeisenberg diamond chain without the secondneighbor interaction between the nodal spins was studied in some detail in our previous work [21], so our particular attention will be primarily devoted to the effect of this interaction term.
Let us at first consider the groundstate phase diagrams of the spin1 IsingHeisenberg diamond chain in zero and nonzero magnetic field. Depending on a relative strength of the secondneighbor interaction only two or three different ground states are available at zero magnetic field: the classical ferrimagnetic (FRI) phase and two quantum antiferromagnetic ones QAF and QAF. The three aforementioned phases can be characterized by the following eigenvectors, the groundstate energy and singlesite magnetizations:

The classical ferrimagnetic phase FRI:
(16) 
The quantum antiferromagnetic phases QAF and QAF:
(17)
The zerofield groundstate phase diagram of the spin1 IsingHeisenberg diamond chain is plotted in Fig. 2(a) in plane for two different values of the secondneighbor interaction . Note that an increase of the Heisenberg coupling strengthens a spin frustration due to a competition between the antiferromagnetic Heisenberg and Ising interactions. If the secondneighbor interaction between the nodal spins is sufficiently small (e.g. ), then, one passes from FRI ground state through QAF ground state up to QAF ground state as the frustration parameter strengthens. On the other hand, QAF ground state without translationally broken symmetry is totally absent in the groundstate phase diagram for strong enough secondneighbor interaction (e.g. ).
(a)  (b) 
(c)  (d) 
In a presence of the external magnetic field one may additionally find another five ground states of the spin1 IsingHeisenberg diamond chain: three quantum ferromagnetic (QFO, QFO, QFO), the unsaturated ferromagnetic (UFM) and the saturated paramagnetic (SPA), which can be characterized through the following eigenvectors, the groundstate energy and singlesite magnetizations:

The quantum ferromagnetic phases QFO, QFO and QFO:
(18) 
The unsaturated ferromagnetic phase UFM:
(19) 
The saturated paramagnetic phase SPA:
(20)
The groundstate phase diagram of the spin1 IsingHeisenberg diamond chain in plane is plotted in Fig. 2(b)(d) for the isotropic Heisenberg coupling and several values of the secondneighbor interaction . It can be seen from Fig. 2(b) that the secondneighbor interaction between the nodal spins gives rise to two new ground states QAF and UFM, which are absent in the model without this interaction term. In general, the parameter space inherent to the ground states QAF and UFM extends upon rising the secondneighbor interaction as evidenced by from Fig. 2(b) and (c). However, the third novel ground state QFO appears at moderate values of the Heisenberg coupling and magnetic field as far as the secondneighbor interaction becomes sufficiently strong (see Fig. 2(d)).
Now, let us proceed to a comprehensive analysis of the magnetization process at zero as well as nonzero temperatures. To illustrate an influence of the secondneighbor coupling on a magnetization process, Figs. 3(a) and 4(a) compare two different magnetization scenarios of the spin1 IsingHeisenberg diamond chain with and without this interaction term. Fig. 3(a) displays the magnetization curve with a single onethird plateau due to a fieldinduced transition FRISPA for along with the magnetization curve with two successive onethird and twothirds plateaus, which emerge due to two subsequent fieldinduced transitions FRIUFMSPA for . Interestingly, the simple magnetization curve with a single onethird plateau due to a fieldinduced transition FRISPA for may also change to a more complex magnetization curve with three successive plateaus at zero, onethird and twothirds of the saturation magnetization, which emerge due to three subsequent fieldinduced transitions QAFFRIUFMSPA for . It is worthwhile to recall that actual magnetization plateaus and magnetization jumps only appear at zero temperature, while increasing temperature gradually smoothens the magnetization curves. Typical thermal variations of the total magnetization are plotted Figs. 3(b) and 4(b), which evidence pronounced lowtemperature variations of the total magnetization when the magnetic field is fixed slightly below or above the relevant critical field.
(a)  (b) 

(a)  (b) 

Last but not least, let us briefly comment on thermal variations of the zerofield specific heat, which are quite typical for three available zerofield ground states FRI, QAF and QAF. Temperature dependencies of the zerofield specific heat pertinent to the classical FRI ground state are depicted in Fig. 5(a). It can be seen from this figure that the specific heat may display a more peculiar doublepeak temperature dependence in addition to a standard temperature dependence with a single round maximum. The doublepeak temperature dependencies of the specific heat are also quite typical for another ground state QAF, which appears in a rather restricted region of parameter space (see Fig. 5(b)). On assumption that the QAF phase constitutes the ground state thermal variations of the zerofield specific heat with a single or doublepeak structure emerge for greater or smaller values of the Heisenberg interaction as depicted in Fig. 5(c). It could be concluded that the doublepeak temperature dependencies of the specific heat appear in all three aforementioned cases owing to lowlying thermal excitations and consequently, the lowtemperature peak can be always identified as the Schottkytype maximum.
(a)  (b) 

(c)  
4 Conclusion
In the present work, we have examined the ground state, magnetization process and specific heat of the exactly solved spin IsingHeisenberg diamond chain with the secondneighbor interaction between the nodal spins. It has been demonstrated that the considered furtherneighbor interaction gives rise to three novel ground states, which cannot be in principle detected in the simplified version of the spin IsingHeisenberg diamond chain without this interaction term [21]. It should be pointed out, moreover, that the spin IsingHeisenberg diamond chain supplemented with the secondneighbor interaction between the nodal spins does not exhibit more intermediate magnetization plateaus than its simplified version without this interaction term even though all three novel ground states have translationally broken symmetry. This finding seems to be quite surprising, because one could generally expect according to OshikawaYamanakaAffleck rule [26, 27] appearance of new intermediate plateaus at onesixth and fivesixths of the saturation magnetization in addition to the observed intermediate plateaus at zero, onethird and twothirds of the saturation magnetization. Of course, one cannot definitely rule out that the onesixth and/or fivesixths plateaus indeed occur in a zerotemperature magnetization curve of a more general spin IsingHeisenberg diamond chain, which could for instance take into account asymmetric interactions, singleion anisotropy, fourspin and/or biquadratic interactions besides the secondneighbor interaction between the nodal spins. This conjecture might represent challenging task for future investigations.
Acknowledgments
J S acknowledges financial support provided by The Ministry of Education, Science, Research, and Sport of the Slovak Republic under Contract Nos. VEGA 1/0234/12 and VEGA 1/0331/15 and by grants from the Slovak Research and Development Agency under Contract Nos. APVV009712 and APVV140073. N A acknowledges financial support by the MCIRSES no. 612707 (DIONICOS) under FP7PEOPLE2013 and research project no. SCS 15T1C114 grants.
References
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